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\chapter{Comparison of different LMS signal segmentation approaches}


This chapter presents the comparison results for different LMS signal segmentation approaches.

As presented in a previous chapter LMS signal segmentation can be achieved by different correlation structure analysis, so it is essential to compare different correlation structures before introducing 'Minimum State Length (MinStateLength)' feature. Discuss the impact of MinStateLength feature on the previously calculated LMS signal segment.
Below we briefly discuss different correlation structures as introduced in section ~\ref{chap:Correlation Structures of LMS Received Signal}.


\begin{itemize}
\item Correlation Coefficient Matrix (CCM): CCM provides the maximum normalized correlation coefficients between each position in a received signal to all other positions in the same received signal.

Below equation defines CCM
	
	\begin{equation}
	\begin{aligned}
	CCM(t,t+\tau) = \frac{h(t)*h(t+\tau)}{\max{(h(t)^{2}, h(t+\tau)^{2})}}
	\end{aligned}
	\end{equation}
                                                                  
where, $\bm{h}$ is the $ N \times 1$ received signal vector, $CCM$ is the $N \times N $ correlation coefficient matrix, $h(t)$ is the received signal impulse response at position $t$, $h(t+\tau)$ is the received signal impulse response at position $(t+\tau)$ and $ t = 1,2.....(N-1)$.

CCM index 1 means strict stationarity and CCM index 0 represents non-stationarity. 

\item Correlation Matrix Distance (CMD): CMD is dissimilarity measure between two local correlation matrices at different positions on LMS received signal.

\begin{equation}
CMD(\bm{R}(k)\bm{R}(l)) = 1 - \frac{\tr\{\bm{R}(k)\bm{R}(l)\}}{\|\bm{R}(k)\|_{F}\|\bm{R}(l)\|_{F}}
\end{equation}

where, $\bm{R}(k)$ is the $ M \times M $ correlation matrix at position $k$, $\bm{R}(l)$ is the $ M \times M $ correlation matrix at $l$ position, M is window length and $R,l = 1,2,..N$.

CMD index 0 indicates strict stationarity and CMD index 1 means non-stationarity.

For comparison purpose use $1 - CMD(\bm{R}(k)\bm{R}(l))$, and now CMD value 1 represents strict stationarity and 0 indicates non-stationarity.


\item Normalized Correlation Matrix Distance (NCMD): - NCMD is normalized dissimilarity measure between two local correlation matrices at different locations on LMS received signal.

\begin{equation}
NCMD(\bm{R}(k)\bm{R}(l)) =  \frac{CMD(\bm{R}(k)\bm{R}(l))}{K_N}
\end{equation}

Normalization factor $K_N$ for CMD is defined by

\begin{equation}
K_N = 1 - \frac{\min{\{\bm{\Lambda}(k)\}}}{\sqrt{\sum_{j=1}^n \lambda_j^2(k)}}
\end{equation} 

where, $\bm{R}(k)$ is the $ M \times M $ correlation matrix at position $k$, $\bm{R}(l)$ is the $ M \times M $ correlation matrix at $l$ position, $k,l = 1...N$, M is window length and $\bm{\Lambda}(k)$ is the eigenvalue matrix of the local correlation matrix. 

\end{itemize}

The NCMD provides the enhancement on CMD for MIMO channel matrix. Results of NCMD for LMS received signal is same as CMD approach.

Here, we discuss the pros and cons of CCM and CMD approach, as well as working with the figure.


\section{Comparison of correlation approaches without 'MinStateLength'}


% figure Comparison of correlation approaches without 'MinStateLength'

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/cmd_corr_compare06.pdf}
\end{center}
\caption{Comparison of accuracy of CCM and CMD at higher threshold}
\label{fig:Comparison of accuracy of CCM and CMD at higher threshold}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/cmd_corr_compare02.pdf}
\end{center}
\caption{Comparison of accuracy of CCM and CMD at lower threshold}
\label{fig:Comparison of accuracy of CCM and CMD at lower threshold}
\end{figure}

Table ~\ref{tab:Comparison of correlation approaches without 'MinStateLength'} shows a comparison of LMS signal segmentation schemes without MinStateLength.


% table Comparison of correlation approaches without 'MinStateLength'

\begin{center}\label{tab:Comparison of correlation approaches without 'MinStateLength'}
\begin{longtable}{ | p{3cm} | p{5cm} | p{5cm} |}
\caption{Comparison of correlation approaches without 'MinStateLength'}\\
\hline
\textbf{Aspects} & \textbf{CCM} & \textbf{CMD} \\
\hline
\endfirsthead
\multicolumn{3}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
\textbf{Aspects} & \textbf{CCM} & \textbf{CMD} \\
\hline
\endhead
\hline \multicolumn{3}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot


Definition: - & CCM: CCM provides the maximum normalized correlation coefficients between each position in a received signal to all other positions in the same received signal. & CMD: CMD is dissimilarity measure between two local correlation matrices at different positions on LMS received signal.\\ \hline


    Accuracy at higher threshold (strict stationary) \newline e.g. Threshold = 0.6 & CCM calculates the correlation for each snapshot, so on to identify a single point abrupt change exactly as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD at higher threshold}. \newline Chances to cause unnecessary segment borders. & CMD calculates the dissimilarity between local correlation matrices at different positions, so accuracy of this approach depends on the window length (M) as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD at higher threshold}. \newline Experiments show that average window size provides better segmentation ~\ref{fig:Effect of window size on CMD and NCMD}. \\ \hline 
    
    
    Accuracy at a lower threshold (nearer to non-stationary) \newline e.g. Threshold = 0.2 & Lower threshold value (nearer to non-stationarity event) makes possible to reduce stationarity change indicators for LMS signal. \newline By setting the threshold nearer to non-stationary event, CCM may not be able to detect LMS signal structure change precisely as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD at lower threshold}. & CMD may not identify signal structure change with non-stationary threshold as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD at lower threshold}. \newline State change positions are only available with higher threshold (strict stationarity).  \\ \hline
    
    
    Non-stationary region identification & Clear identification of the non-stationarity region is not achieved by CCM as shown in figure ~\ref{Correlation coefficient matrix analysis}. & CMD with moderate window length can provide easy understanding of the non-stationarity area as shown by red color in figure ~\ref{fig:CMD with window length M = 45 for measured data}.  \\ \hline
    
     
    Computational Complexity & CCM expects less computation time than CMD which requires calculation of local correlation matrices. & Computational complexity of the CMD depends on the local correlation matrix dimension, so computational complexity increases with the increment in window length.  \\ \hline
    
    
    Conclusion & CCM performs better nearer to non-stationary threshold. \newline CCM detects unwanted segment borders at higher threshold (strict stationarity), which increases channel simulator complexity. \newline Computational complexity and time is less than CMD. \newline State change indicators accuracy is less compared to CMD. & CMD performs better than CCM for high stationarity area and also presents an easy understanding of the non-stationary region. \newline Computational complexity and time is more compared to CCM. \newline CMD performs better than CCM without MinStateLength. \\ \hline

\end{longtable}
\end{center}




\section{Comparison of correlation approaches with 'MinStateLength'}

This section presents a comparison of correlation structures with MinStateLength parameter. 'MinStateLength' (a number of continuous non-stationary points) comes in effect during the computation of stationarity region indicators.

Table ~\ref{tab:Comparison of correlation approaches with 'MinStateLength'} provides the comparison of segmentation methods in the presence of MinStateLength feature.



% table Comparison of correlation approaches with 'MinStateLength'


\begin{center}\label{tab:Comparison of correlation approaches with 'MinStateLength'}
\begin{longtable}{ | p{3cm} | p{5cm} | p{5cm} |}
\caption{Comparison of correlation approaches with 'MinStateLength'}\\
\hline
\textbf{Aspects} & \textbf{CCM} & \textbf{CMD} \\
\hline
\endfirsthead
\multicolumn{3}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
\textbf{Aspects} & \textbf{CCM} & \textbf{CMD} \\
\hline
\endhead
\hline \multicolumn{3}{r}{\textit{Continued on next page}} \\
\endfoot
\hline
\endlastfoot

Introduction & CCM detects single point quick change in LMS received signal, so complexity of channel simulator increases. \newline After introducing MinStateLength, CCM avoids recognition of the non-stationary area smaller than MinStateLength. & Previously CMD provides better segmentation of signal in terms of stationarity than CCM without MinStateLength. \newline Investigate and compare the effectiveness of CMD with MinStateLength. \\ \hline


    Accuracy at higher threshold (strict stationary) \newline e.g. Threshold = 0.7 & CCM generates state change indicators with accuracy and also avoids unnecessary segment borders with MinStateLength feature as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for strict stationarity with MinStateLength}. & Higher accuracy of stationary segment change indicators is not achieved. \newline CMD Provides some unnecessary segment borders compare to CCM as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for strict stationarity with MinStateLength}. \\ \hline
    
    
    Accuracy at moderate threshold (moderate stationarity) \newline e.g. Threshold = 0.5 & CCM provides essential stationarity segment change indicators as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for moderate stationarity with MinStateLength}. & Stationarity change positions available by CMD may not be accurate as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for moderate stationarity with MinStateLength}. \newline Chance of leaving out some non-stationary events as shown by a red circle in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for moderate stationarity with MinStateLength}. \\ \hline
    
    
    Accuracy at low threshold (nearer to non-stationarity) \newline e.g. Threshold = 0.3 & Possibility of neglecting some non-stationary events, but detects significant changes in LMS signal structure as shown in figure ~\ref{fig:Comparison of accuracy of CCM and CMD for non-stationarity with MinStateLength}. & CMD can not identify stationarity change positions with a threshold value nearer to non-stationary event. Stationarity region borders are not available for CMD as shown by figure ~\ref{fig:Comparison of accuracy of CCM and CMD for non-stationarity with MinStateLength}.\\ \hline
    
    
    Computational Complexity & CCM does not require to calculate of local correlation matrices. \newline Reduction in calculation time is achieved. & Computational complexity of the CMD is depended on the local correlation matrices window length (M), so computational complexity increases with the increment of window length.  \\ \hline
    
    
    Conclusion & CCM performs better for strict stationarity as well as for non-stationary threshold value as shown in figures. \newline MinStateLength feature avoids the problem of unwanted segment identification. \newline CCM requires less computational time than CMD. \newline CCM with MinStateLength generates LMS signal stationarity borders with more precision than CMD and previous approaches. & Performance of CMD with MinStateLength is worst than earlier without MinStateLength approach. \\ \hline
    
    
    Suggestion & CCM with MinStateLength provides the best stationarity segments compared to other approaches discussed previously in this report. \newline CCM with MinStateLength is suggested for segmentation of LMS signal in stationarity area using results shown in this chapter. & CMD without MinStateLength provides better results. CMD can be used without MinStateLength to segment LMS signal in stationarity interval. 

\end{longtable}
\end{center}



% figures Comparison of correlation approaches with 'MinStateLength'

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/cmd_corr_compare_minstatelength07.pdf}
\end{center}
\caption{Comparison of accuracy of CCM and CMD for strict stationarity with MinStateLength}
\label{fig:Comparison of accuracy of CCM and CMD for strict stationarity with MinStateLength}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/cmd_corr_compare_minstatelength05.pdf}
\end{center}
\caption{Comparison of accuracy of CCM and CMD for moderate stationarity with MinStateLength}
\label{fig:Comparison of accuracy of CCM and CMD for moderate stationarity with MinStateLength}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/cmd_corr_compare_minstatelength03.pdf}
\end{center}
\caption{Comparison of accuracy of CCM and CMD for non-stationarity with MinStateLength}
\label{fig:Comparison of accuracy of CCM and CMD for non-stationarity with MinStateLength}
\end{figure}





\section{Results of CCM with MinStateLength}


This section presents results of stationarity change borders of LMS received signal using the correlation coefficient matrix (CCM) with MinStateLength analysis. 

We presented some result for the comparison between the state change indicators generated by the proposed method 'CCM with MinStateLength' and previously generated state sequence. 

Figure ~\ref{fig:Comparison of states between CCM with MinStateLength and pre-defined states for route02} presents an analysis of the state sequence (1 = Good, 2 = Bad) for LMS measurement route with the proposed method 'CCM with MinStateLength'. Here, fixed threshold (-5 dB) is used to divide generated stationarity segment indicators into Good or Bad state.


% figure Comparison of states between CCM with MinStateLength and pre-defined states

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/ccm_predefined_compare_route02.pdf}
\end{center}
\caption{Comparison of states between CCM with MinStateLength and pre-defined states for route02}
\label{fig:Comparison of states between CCM with MinStateLength and pre-defined states for route02}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{./bilder/ccm_predefined_compare_route04.pdf}
\end{center}
\caption{Comparison of states between CCM with MinStateLength and pre-defined states for route04}
\label{fig:Comparison of states between CCM with MinStateLength and pre-defined states for route04}
\end{figure}